Stochastic Volatility with Jumps — SVJ, SVJJ, and the Bates Model

The Heston model captures the long-dated smile but struggles at short maturities (the short-dated skew exceeds the $1/\sqrt{T}$ bound). Pure Lévy models (VG, NIG, Merton) match the short-dated smile but produce a term structure that flattens too quickly (central limit theorem convergence to Gaussian). The natural resolution: combine stochastic volatility with jumps.

The Stochastic Volatility with Jumps (SVJ) model, introduced by Bates (1996), adds a compound Poisson jump component to the Heston SDE. It preserves the Heston Fourier pricing structure — the characteristic function remains in closed form — while generating a

1/T

component in the short-dated skew. The further extension SVJJ (simultaneous jumps in both

S

and

v

) improves the vol-of-vol match at intermediate maturities.

Learning objectives. By the end of this note you should be able to:

state the SVJ (Bates) SDE and write its characteristic function;
explain how adding jumps changes the short-dated skew while the stochastic vol governs the longer-dated term structure;
calibrate the Bates model to a term structure of implied vol smiles.

In this lesson

8.1 The SVJ (Bates) model
8.2 The SVJ characteristic function
8.3 SVJJ — simultaneous jumps in $S$ and $v$
8.4 Term structure decomposition
8.5 Calibration

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